staring contest
Noun
Meaning
a game in which two people maintain eye contact for as long as possible, the loser being the one who is the first to blink or look away.
a confrontation between two parties in which neither side is prepared to back down.
Source: en.wiktionary.orgExamples
I like intelligent women. When you go out, it shouldn't be a staring contest.
Can't be any harder than sitting here and having a staring contest with mortality.